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Monopole production and rapid decay of gauge fields
Aiichi IwazakiNishogakusha University
High energy heavy ion collisions
Generation of color electric and magnetic fieldsaccording to a model of color glass condensate
thermalizedquark gluon plasma
Quarks and gluons are producedby the rapid decay of the gauge fields
decay time< <1fm/c
(~ 0.5fm/c ?)
Hirano, Nara2004
We have not yet foundthe rapid decay mechanismof the gauge fields.
We wish to propose a rapid decay mechanism of the gauge fields.
High energy densityof the color gauge fields
~ 2
4
gQs
width offlux tube
Homogeneous in longitudinal direction
2~)( sQgBgE
field strength
GeVGeVQs 2~1
0 BE
sQ
1 sQ
saturation momentum
Ensemble ofZ2 vortices ?E
B
B
BE
E
EB
EB
The gauge fields are unstable.
Dumitru,NaraPetreska, 2013
( RHIC or LHC )
Characteristics of the gauge fields
|A(pL=p,t)/A(pL=p,t=0|2
x
FF xtBxtBD 222' ))),(()),(((
J. Berges, S. Scheffer and D. Sexty, 2008
Exponential growth of the distancebetween nearby gauge fields at t=0
B B’ t=0
FFD
Kunihiro, Muller, Ohnishi, Schafer, Takahashi, Yamamoto (2010, 2013)
Exponential growth offluctuations aroundthe gauge fields
Exponential growth oflongitudinal pressure offluctuations aroundthe gauge fields in expandingglasma (τ,η coordinates )
Romatschke and Venugopalan 2006
Fukushima and Gelis 2011
It has been found thatthese instabilities do notlead to sufficiently rapid decayof the gauge fields for QGP to berealized within 1fm/c.They have been discussed tobe Nielsen-Olesen instabilities.
Nielsen-Olesen instability
classical instability in SU(2) gauge theory
22
22241
)(4
)(
||2
1)(
4
1
gAAig
DDAAFLgluon
3 AA
2
21
iAA
( Electromagnetic fieldsrepresent the backgroundgauge fields )
3,2,1,)(||2
1|| 22
jiAAigDDL jiijjiijjiit
The term can be positive or negativefor arbitrary magnetic field B0
,
00 Agauge
igAD iii
Iwazaki 2008,
Itakura, Fujii, 2008
( charged vector fieldsare fluctuations aroundthe gauge fields )
Nielsen and Olesen 1978,
Homogeneous B )0,0,0( 1221321 AABBB
gBDL t 2|||| 22
221
i
negative potential
0),exp( 22 sQgBti
Nielsen-Olesen unstable modesoccupying lowest Landau level
Negative potentialfor homogeneous B -2gB
Potential forinhomogeneous B
)(~;0),exp( 22seffeff QOgBgBgBti
growth rate
sQgB
growth rate
effgB
gBD 222
02
),exp( t
gB2Bound states in the Lowest Landau level
Bound states exist with
We may represent these bound states by using effective magnetic field
Eq. of motion
Numerical results ( nonexpanding glasma )
sQgtA )25.0~17.0())(3.0~2.0();exp( 4/12
growth rate
saturationmomentum
cfmQ s
/2.117.0
11 A roughly estimated decay timeof the background fields
5.22
1
4
12
gg
GeVQ
s
s
J. Berges, S. Scheffer and D. Sexty, 2008
Kunihiro, Muller, Ohnishi, Schafer, Takahashi, Yamamoto (2010, 2013)
When we represent the growth rate by using effective homogeneous magnetic field such as ,we find
effgB
effgB2)17.0( seff QgB
effteff gBDL 2|||| 22
222 06.0)17.0(22 sseffeff QQgBm
small effective mass( imaginary )
Effective Lagrangian of describing the instabilityunder inhomogeneous magnetic fields is given such that using effective homogeneous magnetic field
seff QgB 17.0~
effgB
small growth rate
)exp( t
long decay time cfm /2.1~1
Using the effective Lagrangian, we calculate the back reaction of the unstable modes on the background gauge fields and show how fast the fields decay.
We show that the monopole production leads to much morerapid decay of the gauge fields than the production of Nielsen-Olesen unstable modes
Similarly, we wish to calculate the back reaction of magnetic monopoles on the background gauge fields by using an effective Lagrangian of the monopoles.
The monopoles are such objects whose condensation gives rise to “quark confinement” in QCD.
dimi
di
mmmd
mtmonopole
AigD
mDL
42222 ||||||||
Effective Lagrangian of magnetic monopolesdescribing quark confinement
m
di
m
A
gg
GeVOm
2)1(
larger than
magnetic charge
dual gauge fields
GeVQfor
GeVQfor
gg
g
s
s
sm
23
15.2
2
12
‘tHooftMandelstam1976
Koma, Suzuki 2003
GeVmeff 25.0~
describes dual superconductors
GeVQs 1for
monopole field
monopoleL
We calculate the decay time of background colorelectric ( magnetic ) fields by using the effective Lagrangian of Niesen-Olesen unstable modes ( magnetic monopoles ) in expanding glasma(τ,η coordinates )
Note thatthe monopoles occupy Landau levels underbackground electric field, while Nielsen-Olesen modes occupy Landau levels under magnetic field.
Our assumptions
Relevant monopoles occupy only the lowest Landau level
Their distribution is almost homogeneous in transverse plain so that magnetic field affected bythe monopole production is almost homogeneous.
The similar assumptions for Nielsen-Olesenunstable modes are adopted.
)||1
|(| 22
244 effteff gBDdxLdxg
igAD
)|),(||),(|1
|),((| 222
224
effeff gBDddLLdxg
2
0
22
0
|)(|;)(),(
n
TnnTn
Tnn xadxLxa
)4
||()(
2
21Teffn
n
xgBExpixx
Effective Lagrangian of Nilesen-Olesen (N-O) unstable modes in τ,ηcoordinates
)log(2
1,
30
3023
20 xx
xxxx
Assuming homogeneousdistribution of N-O modesin transverse plain, the dynamicalvariable is left.
2
0
|)(|n
Tnn xa
),(
( wave functions ofthe lowest Landau level )
)||||||1
|(| 22222
244mmeffmm
dmeff mEgDdxLdxg
dm
d AigD
)|),(||),(||),(|1
|),((| 22222
22
mmeffmmm mEgDddL 2
0,
22
0, |)(|;)(),(
n
TnmnTn
Tnmnmm xadxLxa
)4
||()(
2
21,Teffmn
nm
xEgExpixx
wave functions ofthe lowest Landau level under
Effective Lagrangian of magnetic monopoles in the lowest Landau level
effmEg
d
m
effmeff
AA
mm
EggB
00
Nielsen-Olesen monopole
Equations of motion of Nielsen-Olesen modes
222
22
0
2
|),(|))((2
2)()1
(
),())((1
1
),()1
(
gAd
ggA
gAgBeff
);(),,(2
),(
gAed i homogeneous in Txand
0
1
)(
effgBgB We assume that
background magnetic field decreases
with the expansion
The equations describe how the electric field
decays via the production of Nielsen-Olesen unstable modes )(
1 AE
Maxwell eq.
initial conditionsWe use the initial conditions given by Dusling, Gelis, and Venugopalan
jiijiij
j
ii
i
ii
CzWeCzWegE
i
atccCzWgE
ee
d
realed
i
,2/,2/)1(4/)1(
,2/,2/)1(4/)1(
202,1
1
,,20
4/)1(
21
)()(1
),(
0..)(2
2),(
),,(2
2/)(),(
2/,2/)1(,2/
),(20
,
iiigEz
zW
Whittaker function 0
)(
,,,,
,,,
jiji
jiji
CCCC
CC
2/122/,2/)1(
2/)1(22/,2/)1(
2/)1(
20|)(||)(|
1),( zWezWe
gEiiii
τ→0
Without taking average of initial values
after obtaining the time evolution of we take the initial value, CC )0,(
),(
τ→0
2011, 2012
That is, we include next toleading order of quantumeffects on the evolutionof the background gaugefields.
2/,2/)1(,2/
),(1
),(
20
,20
iiigEz
zWgE
),()2/)0((1
),())((1
1
),()1
(
222
22
0
2
gEgB
gAgB
eff
eff
Positive energy solutions of the equation with the parameters, )0(0 gEgEgBeff
asτ→0
Whittaker function
2/122/,2/)1(
2/)1(22/,2/)1(
2/)1(
20|)(||)(|
1),( zWezWe
gEiiii
τ→0
This initial condition comes from the average,
jiijiij
j CzWeCzWegE
i ,2/,2/)1(4/)1(
,2/,2/)1(4/)1(
202,1
1 )()(1
),(
2,1,,
22/,2/)1(
2/)1(
2,1,,
22/,2/)1(
2/)1(
20|)(|
|)(|1
),(),(
jjjii
jjjii
CCzWe
CCzWe
gE
with the use of the formulae,
τ→0
τ→0
2/20igEz
For simplicity, we take the simple initial conditions,
Similar procedures of initial conditions even in the case of the magnetic monopoles are assumed.
)0(
)(
Bg
Bg
m
m
)0(
)(
gE
gE
Decay of the electric fieldproducing Nielsen-Olesen modes
fm/c
fm/c
Decay of the magnetic fieldproducing magnetic monopoles
ten times more rapid decay
We should note howthe gauge field rapidlydecays producingthe magnetic monopoles.
GeVQ
GeVmGeV
gEgB
ss 2,3/1
7.0,34.0~
)0()0(
tentative results
0.1fm/c
1 fm/c
|)01.0,(| 0 gE
Initial amplitude of Nielsen-Olesen unstable modes
|)01.0,(| effmm Bg
Initial amplitude of magnetic monopoles
The initial amplitude is 10 times larger than theamplitude of Nielsen-Olesen unstable modes
2/1
22/,2/)1(
2/)1(
22/,2/)1(
2/)1(
20 |)(|
|)(|1
zWe
zWe
gE ii
ii
2/1
22/,2/)2.31(
2/)2.31(
22/,2/)2.31(
2/)2.31(
2 |)(|
|)(|1
zWe
zWe
Bg ii
ii
effm
Pair creations of magnetic monopoles under magnetic fields by Schwinger mechnism
production rate of monopoles
3
12101.3~)
)(exp( 2
2
sseffm
effm andGeVQforBg
Egm
103.2~)exp( eff
eff
gE
gB
2)34.0( GeVgBgE effeff
GeVm 7.0
Tanji and Itakura, 2012
The production rate of the monopolesis about 10 times larger than thatof Nielsen-Olesen unstable modes.
Compare the production rate of the monopoles with that of Nielsen-Olesen unstable modes
production rate of N-O modes
conclusions
We have shown that the gauge fields generated afterhigh energy heavy ion collisions decay much more rapidlyproducing magnetic monopoles than Nielsen-Olesenunstable modes.
Although our calculation does not properly take into account precise initial conditions so that the result ispreliminary, it shows that the role of the magnetic monopoles in the realization of thermalized QGP is important.
time
J. Berges, S. Scheffer and D. Sexty, 2008
Numerical simulations
2|)0,(/),(| tptp LL exponential growth ofthe fluctuations
Kunihiro, Muller, Ohnishi, Schafer, Takahashi, Yamamoto (2010, 2013)
Exponential growth of the distancebetween nearby gauge fields
x
FF xtBxtBD 222' ))),(()),(((
B B’ t=0
t
FFD
Numerical simulations
gBDxtB 2||)),(( 22
( in our notations )